Find the coordinates of the point where the line through (3, 8, 9) and (2, 7, 1) crosses the plane passing through the points (2, 2, 1), (3, 0, 1), and (4, 5, 0).

Equation of line is: x - 2 = y + 4 = z + 5 = λ
General point x = -λ + 3, y = λ + 4, z = 6λ - 5
Equation of plane:
| x - 2 y - 2 z - 1 |
| 1 -2 0 |
| 2 3 -1 | = 0
= (x - 2)(-2 - 0) - (y - 2)(1)( -1 - 0) + (z - 1)(1)(3 - (-4)) = 0
= -2(x - 2) + (y - 2) + 7(z - 1) = 0
= -2x + 4 + y - 2 + 7z - 7 = 0
= -2x + y + 7z - 5 = 0
= 2x - y - 7z + 5 = 0 --- (1)
General point of line is a point on (1), Therefore,
2(-λ + 3) - (λ + 4) - 7(6λ - 5) + 5 = 0
= -2λ + 6 - λ - 4 - 42λ + 35 + 5 = 0
= -45λ + 42 = 0
= λ = 42/45 = 14/15
Point of intersection of line and plane
x = -(14/15) + 3 = 3 - 14/15 = (45 - 14)/15 = 31/15
y = (14/15) + 4 = (60 + 14)/15 = 74/15
z = 6(14/15) - 5 = 84/15 - 5 = (84 - 75)/15 = 9/15 = 3/5
(31/15, 74/15, 3/5)